Proceedings of the American Control Conference
Chicago, Illinois June 2000
Calcium Homeostasis: a Feedback Control Point of View
H. El-Samad, M.Khammashl
Electrical and Computer Engineering
Iowa State University
Ames, Iowa 50011
Abstract
In the biological sciences, the mathematical approach
to studying feedback mechanisms has not been common
despite the abundance of such mechanisms in those systems. In this paper, we will try to analyze the calcium
homoeostatic mechanism in this context. In addition t o
a general modeling of this mechanism, the paper will
also focus on parturient paresis, a common disease associated with the onset of parturition in dairy cows, due
to a large increased demand for calcium.
1 Introduction and Background
The use of feedback for the purpose of regulation is
prevalent in biological systems [l, 2, 31. Although ideas
concerning such regulation have been present for a long
time, their implementation in the study of biological
systems started to raise interest only recently. The
major areas of interest in this context were the modeling and analysis of cardiovascular and respiratory systems in addition to some studies of thermoregulation,
endocrine regulation, gastrointestinal secretions, blood
flow, renal plasma regulation, muscle dynamics and eye
accommodation models among many others. [ 1 , 2 , 3 , 4 ] .
The calcium homeostatic mechanism is addressed in this
paper. A model for this mechanism from a control theory point of view is obtained. This model is then used
to study a disease which affects dairy cows and is generally referred to as parturient paresis, or simply milk
fever. Since milk fever is ultimately a failure of the feedback regulatory mechanism to cope with large calcium
demands, it is a good candidate for study using ideas
from feedback control theory. By viewing the calcium
feedback regulation system as a dynamical system, this
study aims to provide a better explanation for the calcium regulation mechanisms during normal operation
and during failure.
Calcium has a particularly important physiological role.
While calcium salts maintain the integrity of the skeleton structure, intracellular calcium ions play an important role, in the activity of a large number of enzymes
and are also involved in conveying information from the
'The authors would like to acknowledge support by NSF grant
ECS-9457485
0-7803-5519-9100 $10.00 Q 2000 AACC
J. Goff
U.S. Department of Agriculture
National Animal Disease Center
Ames, IA 50010
surface to the interior of the cell. Extracellular calcium
ions are also necessary for neuro-muscular excitability,
blood clotting and hormonal secretions among many
other functions [ 6 ] . For this important biochemical role
to be accomplished, extracellular and intracellular concentrations of calcium should be maintained within a
very narrow range, typically between 8 and 10 mg/dl in
dairy cows [7]. The three major compartments involved
with calcium regulation of the equilibrium (homeostasis) are: the bone, the kidney and the intestine. It
is agreed that calcium homeostasis is achieved by the
constant influx and outflow of calcium from and to the
blood plasma under a tight hormonal control that will
be discussed in this paper. This entire homeostatic
mechanism works on increasing the calcium influx into
the extracellular fluid whenever its calcium ion concentration drops below normal due to some kind of calcium demand. Although dairy cows have a very effective mechanism for regulating the calcium concentration in the blood plasma, this mechanism may fail.
On the day of calving, dairy cows produce around 10
liters of colostrum containing 23g or more of calcium,
approximately 6 times as much calcium as the extracellular calcium pool contains. Most animals adapt to
the onset of lactation. However, some become severely
hypocalcemic, which disrupts nerve and muscle function, resulting in recumbency and the clinical syndrome
referred to as parturient paresis [12]. Usually, milk fever
cows are treated with intravenous calcium injection that
keep them alive until the intestinal and bone mechanisms adapt to the large calcium clearance.
2 Derivation of the Model
In [13], Ramberg et.al describe calcium homeostasis in
terms of controlled, controlling and disturbing signals.
Controlled signals are defined to be the plasma calcium
concentration [Ca], and bone calcium Ma, while the
controlling signals are taken to be the intestinal calcium
absorption, the bone calcium resorption and the renal
calcium reabsorption. The disturbing signals are those
that cause loss of calcium from plasma and take the form
of endogenous fecal calcium, clearance via glomerular
filtration, placental calcium transport to the fetus during pregnancy, calcium deposition into the bone, and
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milk calcium secretion during lactation. In [13], it is
also stated that for short term calcium regulation(hours
to weeks), only control of [Ca], may be considered
since it has higher priority than h f b in that time pe+ K n t e s t i n e , where vbone
riod. Let us define VT =
and Kntestine are the rates (g/day) at which calcium is
transported from bone and through intestinal absorption respectively into the plasma. Based on the conservation of mass, we can express the plasma calcium
concentration as follows:
[Ca], = v:1
/ot
(VT - VCl)d7.
Where vol is the total plasma volume (l), and VCl is
the total calcium clearance through the various avenues
mentioned above. Since the plasma calcium concentration are regulated quite accurately to follow a setpoint, the overall closed-loop system can be described
in Figure 1. It should be pointed out that a physiological mechanism exists for generating this setpoint in
the parathyroid gland. This will be elaborated on later
in this paper. At this point the feedback control law
shown in the figure is not specified and will be explored
in the next section.
model would start by the introduction of an integral
term multiplying the error, resulting in a P I controller.
The transfer function from
[Ca], then becomes:
s/vol
[Gal, Vcl
VClto
s2
+ (Kp/vol)s+ ( K I / ? J O l )
which obviously yields a zero steady-state error to a
step change in Vel. Not only is this model consistent
with the zero steady state error observation, the transient response characteristics of the resulting second order system agree quite well with the transient response
characteristics seen in real data. This can seen from Figure 2 where actual response data is plotted at the same
figure with the simulated response of our second order
system. We can see that the system has an overdamped
response to a step disturbance. The single points in
the plot correspond to the ensemble average of Calcium
plasma concentrations for 18 calving cows over a 10 day
period around the day of parturition.The solid plot corresponds to the simulated response for the second order
model when VClis increased from 20 g/day to 70 g f day
on the day of parturition. The values for K p and K I
were selected to minimize the sum of the squared errors
between the actual data and the model. The closeness
of the fit underscores the fact that actual response can
be approximated quite well by second order dynamics.
Figure 1: Overall closed-loop system for calcium homeostasis
2.1 Shortcomings of Existing Model and Necessity of Integral Control
The work in [13] suggests that the control in the system
is proportional to the error. Indeed the expression for
VT in [13] is given by:
Figure 2: Plot of actual data and simulation result for
second order system
VT = 1770(0.104 - [Ca],)
3 Physiological Basis
If one were to analyze this feedback law, it becomes
quickly apparent that it cannot account for the observation since it could be easily seen that the steady-state
The plausibility of this model cannot be advocated until it is physiologically validated, i.e. a physiological
error to a step change of E in VClis ess =
# 0.
counterpart for the PI block is found. In this quest, we
This implies that a sudden and large change in VCl- as
will seek the simplest possible setup that will yield a
would be the case due to the demands of lactation prior
convincing explanation of this mechanism.
to parturition- will not allow the plasma calcium concentration to recover to its setpoint and a steady-state
3.1 Realizing Integration by Means of Hormones
error will persist. This, however, is contrary to obserWe start by considering a single hormone explanation.
vation. Experimental data on normal animals suggests
Suppose the total calcium into the plasma VT is prothat after a period of transition, [Ca], will always reportional to the concentration of one hormone, say horturn to its setpoint value which was present prior to
mone A whose concentration is denoted by [Hormone
the sudden increase in Calcium demand.In order to obA]. Then, PI feedback could only be explained in this
tain this zero steady-state error, our modification to the
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case when
d
-[HormoneAI
dt
0: (error
d
+ K-error)
dt
However, this relation is not very satisfying for at least
two reasons. First, it suggests that there should exist
two mechanisms for the production of Hormone A. Secondly, the above relation indicates that the mechanism
for producing Hormone A must somehow rely on measurements of the derivative of the error which is likely
to be a difficult and noise prone task. Alternatively, if
one considers two hormones to realize the PI control for
calcium regulation, an elegant and quite plausible solution emerges. Suppose we have two hormones A and B.
If
[HormoneAI 0: error
d
-[HormoneB] 0: [HormoneAI
dt
VT = V A
VB
+
where
VA 0: [HormoneAI
VB 0: [HormoneB]
Then, the proportional component of our PI control is
given by VA,while the integral component is given by
V B . Furthermore, the concentration of Hormone A provides a measure of the error while the concentration of
Hormone B provides a measure of the integral of the error. Without further information, it is hard to say more
about the feedback control realization based on control
theory alone. In the next section we will see that based
on what is known in the physiology literature on calcium homeostasis, the above postulates are indeed very
good representations of reality.
causes removal of bone salts takes place. If high concentrations of PTH persist, a delayed response (hours
to days) takes place due to the activation of the bone
osteoclasts. This process is known as the osteoclastic
bone resorption. It allows the response to PTH continue even beyond what can be handled by osteocytic
osteolysis. However, in most cases short term needs are
met by osteocytic osteolysis. The effect of PTH on the
kidney is to increase tubular reabsorption of calcium
thus reducing calcium loss through urine. Thus the impact of PTH is to increase immediate calcium transfer
into the blood plasma. On the other hand, the main
role of 1,25-DHCC is to stimulate intestinal absorption
through increasing formation of a calcium-binding protein in the intestinal epithelial cell [9]. Thus the plasma
calcium influx is due to the impact of PTH and that
of 1,25-DHCC. This would seem to coincide with our
third postulate in the previous subsection, if we associate 1,25-DHCC hormone with Hormone B.
How then does the rate of production of Hormone B
(1,25-DHCC) be proportional to the concentration of
Hormone A (PTH)? 1,25-DHCC is produced from a biologically inactive form of Vitamin D after it undergoes several hydroxylations steps in the liver and kidney [6, 7,8,9]. The last hydroxilation step in the kidney
takes place only under stimulation of PTH. Therefore,
the PI action is implemented by the mechanisms underlying the production of these 2 hormones.
3.3 Homeostatic System Failure in Milk Fever
Animals
In order to relate the failure of the feedback components to milk fever, we need to look at their physiological sources. AS mentioned before v -= V b o n e + x n t e s t i n e .
For the bone calcium, we can write:
3.2 The Endocrinology of Calcium Homeostasis
h o n e = aboneVB
(1)
It is agreed upon that when calcium demand from the
where VB is the quantity of calcium that is available for
plasma is increased, calcium homeostasis is achieved by
resorption in bone. Clearly, 0 5 (Yoone 5 1. w e know
the influx of calcium from and to the blood through
that PTH stimulates bone resorption , therefore:
bone, kidney, and intestine under the tight control of
two major hormones: Parathyroid Hormone (PTH),
and the most important metabolite of Vitamin D: 125 Dihydroxycholecalciferol (1,25-DHCC) [6, 71. The
When f b ( ’ ) can be adequately described by a linear reparathyroid hormone is produced in the parathyroid
lation, we would have f b ( [ P T H ] )= (Yb[PTH]for some
glands in response to a decrease in the calcium plasma
constant a b . On the other hand, we know that at any
concentration from the desired setpoint. Experiments
given time PTH secretion by the parathyroid gland-and
have shown that the production is very much a linear
hence PTH plasma concentration-is proportional to the
function of the deviation from the setpoint.Thus, PTH
[Ca], deficiency. Thus, the overall relationship would
is an accurate measure of the error at a given time,
be:
which would be in agreement with our first postulate
h o n e = K,.e
in the previous subsection if PTH is taken to correWhere e := r - [Ca],
spond to Hormone A. Now PTH interacts mainly with
the bone and kidney. In the bone, PTH has a marked
Similarly, for intestinal absorption we have
effect on bone: upon the increase in PTH concentration a fast process known as osteocytic osteolysis that
Vintestine = aintestine V,
(3)
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where
is the calcium available in the diet. As beaintenstine 5 l . Since intestinal absorption
fore, 0
is stimulated by concentrations of 1,25-DHCC we shall
model
aintestine = fi([l,25 - DHCC])
(4)
<
R
But when f i ( . ) can be described adequately by a linear
function, we can write f i ( [ l , 25 - DHCC]) = ai[l,25 DHCC] for some constant a i . In this case, we have
The relationship between PTH and the 1,25-DHCC production rate could be written as
d
-[1,25 - DHCC] = a,[PTH]
dt
Lumping the above relationships together, we get:
Figure 3: 1st Figure: A and B: Typical rumen and abomasal pressure recordings prior to the induction
of hypocalcemia in one cow. C and D: Typical
rumen and abomasal pressure recordings at the
end of the induced hypocalcemic state (taken
from [14]). 2nd Figure:Multiplicative reduction
factor reflecting the effect of [Ca], deficiency on
bone resorption and intestinal absorption
t
Kntestine
= KI
edT
(6)
of such a multiplication factor is neither practically feasible nor important. The main point here is to study
the qualitative effects that such a multiplicative factor
may have on the system. The resulting overall nonlinear
model is shown in figure 4.
While this model may be adequate to describe the
hypocalcemic case, it must be modified by including
nonlinear effects inherent to the calcium homeostatic
regulation mechanism in order to explain parturient
paresis. Nonlinear effects that can be added to our
linear model take the form of saturation terms in the
PI control block. The physical interpretation of these
I
U
I
saturations follows from the observation that PTH and
1,25-DHCC quantities are limited by the physiological
capacities of the tissues involved in their production.
Figure 4: Nonlinear closed-loop model
Another key nonlinear effect introduced into the model
reflects the impact of reduced [Ca], on the intestinal absorption processes. In [14], Daniel establishes through
3.4 State-Space Representation and Phase Porexperimentation that there is a highly significant corretraits
lation between plasma calcium level and the amplitude
A state-space description of the system shown in figure 4
and rate of both gut and abomasal motility in cows.
could be given by the following:
This observation is explained in terms of the general
1
effects of a depression of levels of ionized calcium on
XI
= -[satl(Kp(r - 21)) + f(z1)satz(z2)- Vel]
U01
smooth muscle contractility and neuromuscular transj.2
= KI(r-21)
mission. This reduction in motility is in turn translated
into a reduction in intestinal abosrption. Figure 3 shows
where 21 is the output of the calcium pool integrator
the dramatic effect of [Ca], deficiency on rumen and
and 22 is the output of the PI block integrator. f(.) is
abomasal motility. Therefore, the effect of low plasma
the
nonlinear function corresponding to the effect of excalcium on the supply rate of calcium has been modcessive
decrease of [Ca], on the absorption coefficient.
eled as a nonlinear, monotonically increasing function
The
equilibrium
point-of this system can be easily commultiplying the absorption and coefficient and assumputed
to
be:
ing a value of unity at the set point (see Figure 3). This
multiplication factor is a quadratic function of [Ca],
which has been obtained by considering the product of
The phase portraits of the system described above have
the rate and amplitude linear regression equations for
been numerically computed for Vel = 70, Vi = 100
rumen motility given in [14]. This function was then
and r=0.08 . It is interesting to see (Figure 5) that
normalized to have unity value at normal [Ca], level
(considered in the simulation to be O.O8g/l). It should
for K, = 4300 and KI = 1800 (the parameters identified from experimental data for nonmilk fever animals),
be pointed out here that obtaining an accurate shape
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Those nonlinearities are due to the saturations in PTH
and 1,25-DHCC production and to a postulated reduction in calcium intestinal absorption after an excessive
decrease in calcium concentration. therefore, the model
was able to reproduce both normal and abnormal behavior of the calcium homeostatic mechanism.
the post disturbance solution trajectory goes to the new
equilibrium state which correspond to the [Ca], setpoint and the new clearance rate, while for Kp = 2500
6) we see that the solution startand KI = 1200 (figure
. ing at
( og) would go unstable and the regulatory
system 6reaks iown, as would be seen in the case of milk
fever. These phase portraits correspond to the system
operating at its equilibrium (prior to calving) and then
being impacted by a constant disturbance of amplitude
70g/day. This effect can be duplicated for other low
values of Kp and K I , but does not appear for large values of K p and K I . In fact, whether or not breakdown
takes place for a given disturbance level in this model
depends to a great extent on the level of undershoot exhibited by the response of the 2nd order linear system
to the disturbance input, on the saturation limits and
on the function f(.). Indeed, it could be-shown andytically that given K, and K I , instability would occur for
a range of monotonically increasing reduction functions
f ( - )assuming sufficiently small values at the initial conditions. This result will be reported elsewhere.
References
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[2] H. Milhorn, The Application of Control Theory
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[3] D. Eggs, Control Theory and Physiological Feedback Mechanisms, Williams & Willkins, 1970.
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Figure 5: Phase portrait for high values of K p and KI
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-e-----
[lo] J.B. Anderson, “Parturient Hypocalcemia” , Academic Press, New York, 1970.
. .., ” ” ., .. .”
.*
[ l l ] C.R. Curtis, H.N. Erb, C.J. Sniffen, R.D. Smith,
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.. ”
Figure 6: Phase portrait for low values of K p and KI
4 Conclusion
rn this paper, a study of the calcium homeostatic
anism with special reference to milk fever in dairy
was presented. A linear model was developed for the
normal hypocalcemic case at parturition. The integral
action of the model as well as the proportional action
have a physiological basis and can be explained in terms
of the interactions of two hormones. Nonlinearities were
then added to account for the milk fever recumbency.
[12] G.R. Oeztel, J.P. Goff, “Milk Fever (Parturient
Paresis) in Cows, Ewes and Doe Goats”, In Current Veterniary Therapy, J.Howard (ed.), W.B. Saunders CO,
Philadelphia, 1998.
[13] C.F. Ramberg, E.K. Johnson, R.D. Fargo, D.S.
Kronfeld, “ Calcium Homeostasis in Cows , with Special
R&erence to Parturient Hypocalcemia”, Am. J. PhysPP. R698-R7047
1984
iol.,
[14] R.C.W. Daniel, “Motility of Rumen and Abomasum during Hypocalcemia”, Can. J. Comp. Med.,
Vo1.47, pp. 276-280, 1983.
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